We're gunning for the area of this region here: ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_graphik_9.png)
Let's find the area inside the graph r = 2cos θ and subtract the area inside the graph r = cos θ. The area inside r = 2cos θ is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_33.png)
and the area inside r = cos θ is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_34.png)
So the area of the region in between the two graphs is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_35.png)
Since the limits of integration on the two integrals are the same, we can combine them into the single integral ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_36.png)
We could also find this area one slice at a time. That is, we could find the area of the region between the graphs r = 2cos θ and r = cos θ by slicing the larger region into pizza slices, figuring out the area of the "crust" on each slice, and adding those areas up. The area of the full slice is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_37.png)
and the area of the juicy center part is ,
so the area of the crust is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_39.png)
This simplifies to ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_40.png)
When we add up all all the crust areas and let the number of pieces approach ∞, we get the integral ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_41.png)
To generalize with a nice, neat, pizza-making formula, when we have a graph router and a graph rinner, the area in between the graphs for α ≤ θ ≤ β is ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_42.png)
Be Careful: When working with two different radii, we don't want the formula below, ![](https://media1.shmoop.com/images/calculus/calc_arvolarclen_polarcoord_latek_43.png)
for the area between the graphs router and rinner. This may look less complicated, but it's wrong. The two radii must be squared separately and then subtracted. |